Assessing distributional assumptions using the nullabor package
In nullabor: Tools for Graphical Inference

Assessing distributional assumptions using the nullabor package

library(knitr)
opts_chunk$set(out.extra='style="display:block; margin: auto"', fig.align="center")

The nullabor package provides functions to visually assess distributional assumptions.

library(nullabor)

Start by specifying the distribution family under the null hypothesis. The options available are:

Beta distribution: beta
Cauchy distribution: cauchy
$\chi^2$ distribution: chi-squared
Gamma distribution: gamma
Geometric distribution: geometric
Lognormal distribution: lognormal
Logistic distribution: logistic
Negative binomial distribution: negative binomial
Binomial distribution: binomial
Normal distribution: normal
Poisson distribution: poisson
Student's t distribution: t
Uniform distribution: uniform
Weibull distribution: weibull

You can also specify the parameters of the distribution. This is required for uniform, beta, and binomial distributions.

Using histograms

The first option is to use histograms with kernel density estimates.

To test the hypothesis that the variable total_bill in the tips dataset follows a normal distribution, we draw a histogram lineup plot using lineup_histograms as follows:

data(tips)
lineup_histograms(tips, "total_bill", dist = "normal")

Run the decrypt code printed in the Console to see which plot belongs to the tips data.

To instead test the hypothesis that the data follow a gamma distribution, we use dist = "gamma":

lineup_histograms(tips, "total_bill", dist = "gamma")

Specifying distribution parameters

In some cases, we need (or want) to specify the entire distribution, and not just the family. We then provide the distribution parameters, using the standard format for the distribution (i.e. the same used by r*, d*, p*, and q* functions, where * is the distribution name).

As an example, let's say that we want to test whether a dataset comes from a uniform $U(0,1)$ distribution. First, we generate two example variables. x1 is $U(0,1)$, but x2 is not.

example_data <- data.frame(x1 = runif(100, 0, 1),
                           x2 = rbeta(100, 1/2, 1/2))

For the uniform distribution, the parameters are min and max (see ?dunif). To test whether the x1 data come from a $U(0,1)$ distribution, we specify the distribution parameters as follows:

lineup_histograms(example_data, "x1", dist = "uniform", params = list(min = 0, max = 1))

And for x2:

lineup_histograms(example_data, "x2", dist = "uniform", params = list(min = 0, max = 1))

Using Q-Q plots

An alternative to histograms is to use quantile-quantile plots, in which the theoretical quantiles of the distribution are compared to the empirical quantiles from the (standardized) data. Under the null hypothesis, the points should lie along the reference line. However, some deviations in the tails are usually expected. A lineup plot is useful to see how much points can deviate from the reference line under the null hypothesis.

To create a Q-Q lineup plot using the normal distribution as the null distribution, use lineup_qq as follows:

lineup_qq(tips, "total_bill", dist = "normal")

Again, some distributions require parameters to be specified. This is done analogously to how we did it for histograms:

lineup_qq(example_data, "x1", dist = "uniform", params = list(min = 0, max = 1))

Changing plot appearance

For both histograms and Q-Q plots, you can style the plot using arguments for color and opacity, as well as using ggplot2 functions like theme.

Histograms:

library(ggplot2)
lineup_histograms(example_data, "x1",
                  dist = "uniform",
                  params = list(min = 0, max = 1),
                  color_bars = "white",
                  fill_bars = "#416B4B",
                  color_lines = "#7D5AAD"
                  ) + theme_minimal()

Q-Q plots:

lineup_qq(tips, "total_bill",
          dist = "gamma",
          color_points = "cyan",
          color_lines = "#ED11B7",
          alpha_points = 0.25) +
  theme_minimal() + 
  theme(panel.background = element_rect(fill = "navy"),
        axis.title = element_text(family = "mono", size = 14))

References

Buja, A., Cook, D., Hofmann, H., Lawrence, M., Lee, E.-K., Swayne, D. F, Wickham, H. (2009) Statistical Inference for Exploratory Data Analysis and Model Diagnostics, Royal Society Philosophical Transactions A, 367:4361--4383, DOI: 10.1098/rsta.2009.0120.